AF: Small: Expanding the Reach of Topological Data Analysis

Ohio State University, The, Columbus OH

Investigators

Abstract

Data science and analysis have taken a center stage in modern years for a swath of applications embracing daily life. In general, AI and machine learning have been the backbone of this development. Recently, a technique based on the classical mathematical area of topology has evolved that is showing promise to complement and in some cases be an alternative to AI techniques. As this emergent area, known as topological data analysis (TDA), expands its reach, the task of extracting intelligent summaries out of diverse, complex data becomes increasingly challenging. To meet this challenge, TDA also needs to expand its current repertoire of techniques by making further and deeper connections between mathematical and algorithmic concepts. This project aims to develop these techniques while training graduate students and other workforce in data science as a consequence. The project, in particular, will investigate several novel mathematical concepts in conjunction with computations to address various challenges appearing in the following topics: (i) augmenting topological summarization with geometry, (ii) connecting discrete Morse theory to further applications, (iii) developing persistence theory for combinatorial dynamical systems, and (iv) developing multi-parameter persistence from a computational view-point. A successful algorithmic theory supported by sound mathematics for the four aforementioned themes can provide a powerful tool for data exploration and analysis in a spectrum of engineering and scientific fields. The proposed outreach activities will also further consolidate the ongoing effort of bringing together different communities in science and engineering engaged into modern data analysis. The project will help train graduate students who will develop skills in mathematics and theoretical computer science, most notably in algorithms and topology, in writing efficient and usable software, and in its application to analyzing real-world data sets. Efforts will be made to recruit and mentor students from under-represented groups, as has already been done by the investigator. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.